Cylindrical multipole moments
Cylindrical multipole moments are the coefficients in a series expansion of a potential that varies logarithmically with the distance to a source, i.e., as . Such potentials arise in the electric potential of long line charges, and the analogous sources for the magnetic potential and gravitational potential.
For clarity, we illustrate the expansion for a single line charge, then generalize to an arbitrary distribution of line charges. Through this article, the primed coordinates such as refer to the position of the line charge(s), whereas the unprimed coordinates such as refer to the point at which the potential is being observed. We use cylindrical coordinates throughout, e.g., an arbitrary vector has coordinates where is the radius from the axis, is the azimuthal angle and is the normal Cartesian coordinate. By assumption, the line charges are infinitely long and aligned with the axis.
Cylindrical multipole moments of a line charge
The electric potential of a line charge located at is given by
where is the shortest distance between the line charge and the observation point.
By symmetry, the electric potential of an infinite linecharge has no -dependence. The line charge is the charge per unit length in the -direction, and has units of (charge/length). If the radius of the observation point is greater than the radius of the line charge, we may factor out
and expand the logarithms in powers of
which may be written as
where the multipole moments are defined as
Conversely, if the radius of the observation point is less than the radius of the line charge, we may factor out and expand the logarithms in powers of
which may be written as
where the interior multipole moments are defined as
General cylindrical multipole moments
The generalization to an arbitrary distribution of line charges is straightforward. The functional form is the same
and the moments can be written
Note that the represents the line charge per unit area in the plane.
Interior cylindrical multipole moments
Similarly, the interior cylindrical multipole expansion has the functional form
where the moments are defined
Interaction energies of cylindrical multipoles
A simple formula for the interaction energy of cylindrical multipoles (charge density 1) with a second charge density can be derived. Let be the second charge density, and define as its integral over z
The electrostatic energy is given by the integral of the charge multiplied by the potential due to the cylindrical multipoles
If the cylindrical multipoles are exterior, this equation becomes
where , and are the cylindrical multipole moments of charge distribution 1. This energy formula can be reduced to a remarkably simple form
where and are the interior cylindrical multipoles of the second charge density.
The analogous formula holds if charge density 1 is composed of interior cylindrical multipoles
where and are the interior cylindrical multipole moments of charge distribution 1, and and are the exterior cylindrical multipoles of the second charge density.
As an example, these formulae could be used to determine the interaction energy of a small protein in the electrostatic field of a double-stranded DNA molecule; the latter is relatively straight and bears a constant linear charge density due to the phosphate groups of its backbone.